The resolution of the Gibbs phenomenon for spherical harmonics

Anne Gelb

Research output: Contribution to journalArticlepeer-review

40 Scopus citations

Abstract

Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere. Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first N expansion cc efficients (either Fourier or Gegenbauer) of a piecewise analytic function f(cursive Greek chi) is enough to recover an exponentially convergent approximation to the point values of f(cursive Greek chi) in any subinterval in which the function is analytic. Here we take a similar approach, proving that knowledge of the first N spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function f(θ, φ) in any subinterval [θ1, θ2], φ; ∈ [0, 2π], where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.

Original languageEnglish (US)
Pages (from-to)699-717
Number of pages19
JournalMathematics of Computation
Volume66
Issue number218
DOIs
StatePublished - Apr 1997

Keywords

  • Exponential accuracy
  • Gegenbauer polynomials
  • Gibbs phenomenon
  • Spherical harmonics

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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